This a new matrix inversion test written from scratch to be suitable for running in CI: quick to run and automatically detects success/failure. This all is a result of what I learnt while working on https://gitlab.freedesktop.org/pq/fourbyfour Computing the residual error with infinity norm comes straight from fourbyfour documentation on how to evaluate matrix inversion error. Most of the hard-coded test matrices have been generated with fourbyfour project as well, as it contains the generator code. The matrices are hard-coded here also to make testing faster, but primarily because the generator code needs BLAS and LAPACK, and having those as Weston dependencies would be far too much just for this. Now, if someone wants to modify weston_matrix stuff, we should at least detect matrix inversion and multiplication bugs. Signed-off-by: Pekka Paalanen <pekka.paalanen@collabora.com>dev
Pekka Paalanen
3 years ago
committed by
Pekka Paalanen
parent
8bbd1a995b
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892421a93e
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/*
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* Copyright 2022 Collabora, Ltd. |
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* |
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* Permission is hereby granted, free of charge, to any person obtaining |
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* a copy of this software and associated documentation files (the |
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* "Software"), to deal in the Software without restriction, including |
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* without limitation the rights to use, copy, modify, merge, publish, |
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* distribute, sublicense, and/or sell copies of the Software, and to |
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* permit persons to whom the Software is furnished to do so, subject to |
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* the following conditions: |
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* |
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* The above copyright notice and this permission notice (including the |
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* next paragraph) shall be included in all copies or substantial |
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* portions of the Software. |
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* |
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* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, |
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* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF |
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* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND |
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* NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS |
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* BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN |
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* ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN |
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* CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE |
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* SOFTWARE. |
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*/ |
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#include "config.h" |
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#include <math.h> |
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#include <libweston/matrix.h> |
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#include "weston-test-client-helper.h" |
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/*
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* A helper to lay out a matrix in the natural writing order in code |
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* instead of needing to transpose in your mind every time you read it. |
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* The matrix is laid out as written: |
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* ⎡ a11 a12 a13 a14 ⎤ |
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* ⎢ a21 a22 a23 a24 ⎥ |
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* ⎢ a31 a32 a33 a34 ⎥ |
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* ⎣ a41 a42 a43 a44 ⎦ |
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* where the first digit is row and the second digit is column. |
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* |
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* The type field is set to the most pessimistic case possible so that if |
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* weston_matrix_invert() ever gets special-case code paths, we don't take |
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* them. |
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*/ |
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#define MAT(a11, a12, a13, a14, \ |
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a21, a22, a23, a24, \
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a31, a32, a33, a34, \
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a41, a42, a43, a44) ((struct weston_matrix) \
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{ \
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.d[0] = a11, .d[4] = a12, .d[ 8] = a13, .d[12] = a14, \
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.d[1] = a21, .d[5] = a22, .d[ 9] = a23, .d[13] = a24, \
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.d[2] = a31, .d[6] = a32, .d[10] = a33, .d[14] = a34, \
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.d[3] = a41, .d[7] = a42, .d[11] = a43, .d[15] = a44, \
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.type = WESTON_MATRIX_TRANSFORM_TRANSLATE | \
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WESTON_MATRIX_TRANSFORM_SCALE | \
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WESTON_MATRIX_TRANSFORM_ROTATE | \
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WESTON_MATRIX_TRANSFORM_OTHER, \
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}) |
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static const struct weston_matrix IDENTITY = |
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MAT(1, 0, 0, 0, |
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0, 1, 0, 0, |
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0, 0, 1, 0, |
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0, 0, 0, 1); |
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static void |
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subtract_matrix(struct weston_matrix *from, const struct weston_matrix *what) |
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{ |
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unsigned i; |
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for (i = 0; i < ARRAY_LENGTH(from->d); i++) |
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from->d[i] -= what->d[i]; |
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} |
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static void |
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print_matrix(const struct weston_matrix *m) |
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{ |
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unsigned r, c; |
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for (r = 0; r < 4; ++r) { |
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for (c = 0; c < 4; ++c) |
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testlog(" %14.6e", m->d[r + c * 4]); |
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testlog("\n"); |
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} |
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} |
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/*
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* Matrix infinity norm |
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* http://www.netlib.org/lapack/lug/node75.html
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*/ |
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static double |
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matrix_inf_norm(const struct weston_matrix *mat) |
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{ |
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unsigned row; |
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double infnorm = -1.0; |
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for (row = 0; row < 4; row++) { |
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unsigned col; |
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double sum = 0.0; |
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for (col = 0; col < 4; col++) |
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sum += fabs(mat->d[col * 4 + row]); |
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if (infnorm < sum) |
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infnorm = sum; |
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} |
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return infnorm; |
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} |
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struct test_matrix { |
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/* the matrix to test */ |
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struct weston_matrix M; |
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/*
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* Residual error limit; inf norm(M * inv(M) - I) < err_limit |
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* The residual error as calculated here represents the relative |
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* error added by transforming a vector with inv(M). |
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* |
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* Since weston_matrix stores the inverse matrix in 32-bit floats, |
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* that limits the precision considerably. |
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*/ |
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double err_limit; |
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}; |
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static const struct test_matrix matrices[] = { |
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/* A very trivial case. */ |
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{ |
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.M = MAT(1, 0, 0, 0, |
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0, 2, 0, 0, |
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0, 0, 3, 0, |
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0, 0, 0, 4), |
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.err_limit = 0.0, |
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}, |
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/*
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* A very likely case in a compositor, being a matrix applying |
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* just a translation. Surprisingly, fourbyfour-analyze says: |
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* |
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* ------------------------------------------------------------------- |
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* $ ./fourbyfour-analyse 1 0 0 1980 0 1 0 1080 |
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* Your input matrix A is |
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* 1 0 0 1980 |
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* 0 1 0 1080 |
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* 0 0 1 0 |
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* 0 0 0 1 |
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* |
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* The singular values of A are: 2255.39, 1, 1, 0.000443382 |
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* The condition number according to 2-norm of A is 5.087e+06. |
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* |
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* This means that if you were to solve the linear system Ax=b for vector x, |
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* in the worst case you would lose 6.7 digits (22.3 bits) of precision. |
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* The condition number is how much errors in vector b would be amplified |
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* when solving x even with infinite computational precision. |
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* |
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* Compare this to the precision of vectors b and x: |
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* |
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* - Single precision floating point has 7.2 digits (24 bits) of precision, |
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* leaving your result with no correct digits. |
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* Single precision, matrix A has rank 3 which means that the solution space |
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* for x has 1 dimension and therefore has many solutions. |
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* |
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* - Double precision floating point has 16.0 digits (53 bits) of precision, |
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* leaving your result with 9.2 correct digits (30 correct bits). |
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* Double precision, matrix A has full rank which means the solution x is |
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* unique. |
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* |
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* NOTE! The above gives you only an upper limit on errors. |
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* If the upper limit is low, you can be confident of your computations. But, |
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* if the upper limit is high, it does not necessarily imply that your |
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* computations will be doomed. |
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* ------------------------------------------------------------------- |
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* |
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* This is one example where the condition number is highly pessimistic, |
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* while the actual inversion results in no error at all. |
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* |
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* https://gitlab.freedesktop.org/pq/fourbyfour
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*/ |
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{ |
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.M = MAT(1, 0, 0, 1980, |
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0, 1, 0, 1080, |
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0, 0, 1, 0, |
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0, 0, 0, 1), |
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.err_limit = 0.0, |
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}, |
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/*
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* The following matrices have been generated with |
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* fourbyfour-generate using parameters out of a hat as listed below. |
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* |
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* If you want to verify the matrices in Octave, type this: |
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* M = [ <paste the series of numbers> ] |
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* mat = reshape(M, 4, 4) |
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* det(mat) |
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* cond(mat) |
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*/ |
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/* cond = 1e3 */ |
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{ |
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.M = MAT(-4.12798022231678357619e-02, -7.93301899046665176529e-02, 2.49367040174418935772e-01, -2.22400462135059429070e-01, |
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2.02416121867255743849e-01, -2.25754422240346010187e-02, -2.91283152417864787953e-01, 1.49354988316431153139e-01, |
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6.18473094065821293874e-01, 5.81511312950217934548e-02, -1.18363610818063924590e+00, 8.00087538947595322547e-01, |
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1.25723127083294305972e-01, 7.72723720984487272290e-02, -3.76023220287807879991e-01, 2.82473279931768073148e-01), |
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.err_limit = 1e-5, |
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}, |
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/* cond = 1e3, abs det = 15 */ |
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{ |
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.M = MAT(6.84154939885726509630e+00, -6.87241565273813304060e+00, -2.56772939909334070308e+01, -2.52185055099662420730e+01, |
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2.04511561406330022450e+00, -3.67551043874248994925e+00, -1.96421641406619129633e+00, -2.40644091603848320204e+00, |
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5.83631095663641819016e+00, -9.31051765621826277197e+00, -1.80402129629135217215e+01, -1.78475057662460052654e+01, |
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-9.88588496379959025262e+00, 1.49790516545410774540e+01, 2.64975800675967363418e+01, 2.65795891678410747261e+01), |
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.err_limit = 1e-4, |
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}, |
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/* cond = 700, abs det = 1e-6, invertible regardless of det */ |
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{ |
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.M = MAT(1.32125189257677579449e-03, -1.67411409720826992453e-01, 1.07940907587735196449e-01, -1.22163309792902186057e-01, |
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-5.42113793774764013422e-02, 5.30455105336593901733e-01, -2.59607412684229155175e-01, 4.36480803188117993940e-01, |
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2.88175168292948129939e-03, -1.85262537685181277736e-01, 1.46265858042118279680e-01, -9.41398969709369287662e-02, |
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-2.88900393087768159184e-03, 1.57987202530630227448e-01, -1.20781192010860280450e-01, 8.95194304475115387731e-02), |
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.err_limit = 1e-4, |
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}, |
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/* cond = 1e6, this is a little more challenging */ |
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{ |
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.M = MAT(-4.41851445093878913983e-01, -5.16386185043831491548e-01, 2.86186055948129847160e-01, -5.79440137716940473211e-01, |
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2.49798696238173301154e-01, 2.84965614532234345901e-01, -1.65729639683955931595e-01, 3.12568045963485974248e-01, |
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3.15253213984537428161e-01, 3.71270066781250074328e-01, -2.02675623845341434937e-01, 4.19969870491003371971e-01, |
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5.60818677658178832424e-01, 6.45373659426444201692e-01, -3.68902466471524526082e-01, 7.13785795079988516498e-01), |
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.err_limit = 0.02, |
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}, |
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/* cond = 15, abs det = 1e-9, should be well invertible */ |
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{ |
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.M = MAT(-5.37536200142514660589e-05, 7.92552373388843642288e-03, -3.90554524958281433500e-03, 2.68892064500873568395e-03, |
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-9.72329428437283989350e-03, 8.32075145342783470404e-03, 6.52648485926096092596e-03, 1.06707947887298994737e-03, |
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1.04453728969657322345e-02, -1.03627268579679666927e-02, -3.56835980207569763989e-03, -3.95935925157862422114e-03, |
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5.37160838929722633805e-03, 6.13466744624343262009e-05, -1.23695935407398946090e-04, 8.21231194921675112380e-04), |
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.err_limit = 1e-6, |
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}, |
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}; |
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TEST_P(matrix_inversion_precision, matrices) |
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{ |
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const struct test_matrix *tm = data; |
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struct weston_matrix rr; |
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double err; |
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/* Compute rr = M * inv(M) */ |
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weston_matrix_invert(&rr, &tm->M); |
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weston_matrix_multiply(&rr, &tm->M); |
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/* Residual: subtract identity matrix (expected result) */ |
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subtract_matrix(&rr, &IDENTITY); |
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/*
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* Infinity norm of the residual is our measure. |
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* See https://gitlab.freedesktop.org/pq/fourbyfour/-/blob/master/README.d/precision_testing.md
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*/ |
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err = matrix_inf_norm(&rr); |
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testlog("Residual error %g (%.1f bits precision), limit %g.\n", |
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err, -log2(err), tm->err_limit); |
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if (err > tm->err_limit) { |
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testlog("Error is too high for matrix\n"); |
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print_matrix(&tm->M); |
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assert(0); |
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} |
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} |
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